Properties

Label 1089.3.b.f
Level $1089$
Weight $3$
Character orbit 1089.b
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,3,Mod(485,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.485");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1105425137664.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} + 71x^{4} - 80x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 4) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + (\beta_{5} + \beta_{3}) q^{7} + (\beta_{4} - 5 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 4) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + (\beta_{5} + \beta_{3}) q^{7} + (\beta_{4} - 5 \beta_1) q^{8} + (\beta_{5} + 5 \beta_{3}) q^{10} + \beta_{5} q^{13} + (5 \beta_{7} + 3 \beta_{6}) q^{14} + (4 \beta_{2} + 21) q^{16} + ( - \beta_{4} - 4 \beta_1) q^{17} - 5 \beta_{3} q^{19} + (9 \beta_{7} + 11 \beta_{6}) q^{20} + 5 \beta_{7} q^{23} + ( - 5 \beta_{2} - 6) q^{25} + 3 \beta_{7} q^{26} + ( - \beta_{5} - 15 \beta_{3}) q^{28} - 10 \beta_1 q^{29} + ( - \beta_{2} - 25) q^{31} + 21 \beta_1 q^{32} + (\beta_{2} + 35) q^{34} + (\beta_{4} - 16 \beta_1) q^{35} + ( - 5 \beta_{2} - 15) q^{37} + ( - 10 \beta_{7} - 15 \beta_{6}) q^{38} + ( - 5 \beta_{5} - 31 \beta_{3}) q^{40} + ( - 5 \beta_{4} + 10 \beta_1) q^{41} + ( - 6 \beta_{5} + 5 \beta_{3}) q^{43} + ( - 5 \beta_{5} - 10 \beta_{3}) q^{46} + (5 \beta_{7} - 10 \beta_{6}) q^{47} + ( - \beta_{2} + 24) q^{49} + (5 \beta_{4} - 31 \beta_1) q^{50} + (\beta_{5} - 6 \beta_{3}) q^{52} + ( - 15 \beta_{7} + 5 \beta_{6}) q^{53} + ( - 13 \beta_{7} - 33 \beta_{6}) q^{56} + (10 \beta_{2} + 80) q^{58} - 2 \beta_{6} q^{59} + (5 \beta_{5} - 20 \beta_{3}) q^{61} + (\beta_{4} - 30 \beta_1) q^{62} + ( - 5 \beta_{2} - 84) q^{64} - 6 \beta_1 q^{65} + ( - 10 \beta_{2} + 20) q^{67} + ( - 5 \beta_{4} + 24 \beta_1) q^{68} + (19 \beta_{2} + 125) q^{70} + (25 \beta_{7} + 8 \beta_{6}) q^{71} + (4 \beta_{5} + 14 \beta_{3}) q^{73} + (5 \beta_{4} - 40 \beta_1) q^{74} + (10 \beta_{5} + 45 \beta_{3}) q^{76} + (5 \beta_{5} - 15 \beta_{3}) q^{79} + ( - 41 \beta_{7} - 49 \beta_{6}) q^{80} + ( - 25 \beta_{2} - 65) q^{82} + ( - 5 \beta_{4} - 16 \beta_1) q^{83} + ( - 4 \beta_{5} - 14 \beta_{3}) q^{85} + ( - 8 \beta_{7} + 15 \beta_{6}) q^{86} + (10 \beta_{7} - 15 \beta_{6}) q^{89} + ( - 6 \beta_{2} + 60) q^{91} + ( - 15 \beta_{7} - 30 \beta_{6}) q^{92} + ( - 5 \beta_{5} + 20 \beta_{3}) q^{94} + ( - 5 \beta_{4} + 50 \beta_1) q^{95} + (15 \beta_{2} - 55) q^{97} + (\beta_{4} + 19 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} + 168 q^{16} - 48 q^{25} - 200 q^{31} + 280 q^{34} - 120 q^{37} + 192 q^{49} + 640 q^{58} - 672 q^{64} + 160 q^{67} + 1000 q^{70} - 520 q^{82} + 480 q^{91} - 440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 71x^{4} - 80x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 19\nu^{4} - 93\nu^{2} + 70 ) / 73 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{6} + 38\nu^{4} - 40\nu^{2} - 371 ) / 73 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{7} - 160\nu^{5} + 714\nu^{3} - 1623\nu ) / 803 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{6} + 206\nu^{4} - 1016\nu^{2} + 782 ) / 73 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{7} + 493\nu^{5} - 3535\nu^{3} + 8248\nu ) / 803 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\nu^{7} - 480\nu^{5} + 2142\nu^{3} - 51\nu ) / 803 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -43\nu^{7} + 591\nu^{5} - 2547\nu^{3} + 118\nu ) / 803 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{7} + 13\beta_{6} - 3\beta_{5} - 15\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 5\beta_{2} - 24\beta _1 + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 105\beta_{7} + 124\beta_{6} - 9\beta_{5} - 42\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 19\beta_{4} + \beta_{2} - 208\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 798\beta_{7} + 937\beta_{6} + 54\beta_{5} + 303\beta_{3} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
2.65356 + 0.707107i
−2.65356 0.707107i
−0.979091 0.707107i
0.979091 + 0.707107i
0.979091 0.707107i
−0.979091 + 0.707107i
−2.65356 + 0.707107i
2.65356 0.707107i
3.75270i 0 −10.0828 7.83670i 0 −8.18030 22.8268i 0 −29.4088
485.2 3.75270i 0 −10.0828 7.83670i 0 8.18030 22.8268i 0 29.4088
485.3 1.38464i 0 2.08276 0.765629i 0 −8.89285 8.42246i 0 −1.06012
485.4 1.38464i 0 2.08276 0.765629i 0 8.89285 8.42246i 0 1.06012
485.5 1.38464i 0 2.08276 0.765629i 0 8.89285 8.42246i 0 1.06012
485.6 1.38464i 0 2.08276 0.765629i 0 −8.89285 8.42246i 0 −1.06012
485.7 3.75270i 0 −10.0828 7.83670i 0 8.18030 22.8268i 0 29.4088
485.8 3.75270i 0 −10.0828 7.83670i 0 −8.18030 22.8268i 0 −29.4088
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.3.b.f 8
3.b odd 2 1 inner 1089.3.b.f 8
11.b odd 2 1 inner 1089.3.b.f 8
33.d even 2 1 inner 1089.3.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.3.b.f 8 1.a even 1 1 trivial
1089.3.b.f 8 3.b odd 2 1 inner
1089.3.b.f 8 11.b odd 2 1 inner
1089.3.b.f 8 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{4} + 16T_{2}^{2} + 27 \) Copy content Toggle raw display
\( T_{7}^{4} - 146T_{7}^{2} + 5292 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16 T^{2} + 27)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 62 T^{2} + 36)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 146 T^{2} + 5292)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 126 T^{2} + 972)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 556 T^{2} + 52272)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 800 T^{2} + 67500)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 950 T^{2} + 202500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1600 T^{2} + 270000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 50 T + 588)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 30 T - 700)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 7300 T^{2} + 13230000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 5696 T^{2} + 484812)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 5150 T^{2} + 2722500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 10350 T^{2} + 9922500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 17150 T^{2} + 73507500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 40 T - 3300)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 23654 T^{2} + \cdots + 127644804)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 7616 T^{2} + 1881792)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 11250 T^{2} + 29767500)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 11356 T^{2} + 15431472)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 13700 T^{2} + 9922500)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 110 T - 5300)^{4} \) Copy content Toggle raw display
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